To solve a complex problem, it can be helpful to consider a simpler, related problem (Stacey & Groves, 2006). This can help students to identify the structure of a method for solving which can then be used to solve the original problem.
In order for students to do this, they must:
- interpret the worded problem
- identify a simpler problem that is relevant to help solve the given problem.
Teachers can explicitly model to students worked examples where this strategy is going to be helpful and efficient (HITS 4: Worked examples).
Understanding this strategy
To model this strategy, teachers should:
- Choose a problem that can be solved in a number of ways, including the use of a simpler, related problem. This enables discussion of efficiency of this problem solving strategy.
- Ask students to read the problem and try to solve it in groups.
- Discuss the use of the strategy 'identifying a simpler, related problem' and ask students how this could be applied for this given problem.
- Demonstrate the use of the strategy, based on students' responses.
- Discuss how the approach used to solve the simpler problem can help to answer the given problem.
This strategy supports the
Mathematics proficiencies Reasoning ("adapt the known to the unknown") and Problem solving ("use mathematics to represent unfamiliar or meaningful situations") (VCAA, n.d.)
Examples of using simple, related problems to solve complex problems
An example showing how this strategy can be used in a Year 8 Maths class is below.
There are twenty people at a party and they each shake hands once. How many handshakes will there be?
Teacher presents problem and leads a discussion about how students might "identify a simpler, related problem" to solve this more complex problem. Discussion questions could include:
- "What is a simpler, related problem that you could solve here?"
- "How might finding the number of handshakes for four people help in finding the number of handshakes for twenty people?".
Students can consider the number of handshakes if there are a small number of people, e.g. four people. This can be done by 'acting it out' or 'drawing a diagram' or 'making a list'. Example strategies are available below.
Classroom scaffolding and discussion
Students are scaffolded to extend their reasoning to the situation where there are twenty people.
The teacher leads a second class discussion about why it is helpful to use this strategy of identifying a simpler problem, and then extending the strategy to determine the number of handshakes for twenty people.
The teacher prompts students for their reasoning, with questions such as:
- "How did you determine the number of handshakes for four people?"
- "What strategy did you use?"
- "How can you use your approach for working out the number of handshakes for four people to determine the number of handshakes for twenty people?"
- "How did looking at the simpler problem help in working out the number of handshakes for twenty people?"
Person A and B only shake hands once.
A common student error is to count one handshake twice, i.e, AB and BA as two handshakes, instead of one.